A) \[\cot x\]
B) \[\tan x\]
C) \[\sin x\]
D) \[\text{cosec}\,\,x\]
Correct Answer: B
Solution :
\[\frac{\sin \,\,(x+y)-2sin\,\,x+sin\,\,(x-y)}{\cos \,\,(x+y)-2\cos x+\cos \,\,(x-y)}\] \[=\frac{2\sin \frac{x+y+x-y}{2}\cdot \cos \frac{x+y-x+y}{2}-2\sin x}{2\cos \frac{(x+y+x-y)}{2}\cdot \cos \frac{(x+y-x+y)}{2}-2\cos x}\] \[=\frac{2\sin x\cos y-2\sin x}{2\cos x\cos y-2\cos x}=\frac{2\sin x(\cos y-1)}{2\cos x(\cos y-1)}\]\[=\frac{\sin x}{\cos x}=\tan x\]
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